For this reason,
it would be desirable to be able to use a local strategy, in the style of FLP, to
prove impossibility results for two other important tasks:
\(k\) -set agreement [1]}, an extension of consensus,
where processes may decide on up to \(k\)  different values,
and \(M\) -renaming [2]},
where processes must pick distinct names from a given namespace of size \(M\) .
Existing impossibility proofs for these tasks
(e.g., [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]})
are based on topological invariant properties of final configurations of a protocol,
which are global in nature,
namely, all final configurations are analyzed together to argue that
there is no protocol for the task.
For consensus,
these configurations are connected,
in the graph-theoretic sense. For set agreement and renaming,
higher-dimensional connectivity properties are proved.
Researchers have wondered why only global impossibility
proofs have been used for these tasks [10]}.
